# Hokkaido Mathematical Journal

## Hokkaido Mathematical Journal, 39 (2010) pp.217-238

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### Abstract

Let $U$ be a bounded symmetric open neighborhood of the origin of $\R^{m+k} \ (k\geqq 1)$. We shall prove a generalization of the Borsuk's antipodal theorem for an admissible mapping $\varphi:\partial\overline{U}\to \R^m$ and the related topic. We shall generalize the theorem for the case of a bounded symmetric open neighborhood $U$ of the origin of an infinite dimensional normed space $\mathbf{E}$. The Borsuk-Ulam theorem shall be studied for the case of a bounded symmetric open neighborhood $U$ of the origin of an infinite dimensional normed space $\mathbf{E}$.

MSC(Primary) 47H10(MSC2000), 55M20(MSC2000), 57R91(MSC2000) fixed point theorem; antipodal point theorem; Vietoris's theorem;